MAC 2311 - Calculus 1 Section 001 - Spring 2018
This page is devoted to Calculus 1 - Section 001 of Spring 2018 at USF.
Time and location of class:
MW: 11:00 am – 12:15 pm in CMC 130
F: 11:00 am – 11:50 am in CHE 303 (peer leading session)
Office hours (CMC 110):
Monday 12:30-1:30 pm
Wednesday 9:45-10:45 am and 2:00-3:30 pm
Contents:
The course in organized in the following way:
- Functions and limits
- Derivatives
- Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions
- Applications of Differentiation
- Integrals
Textbook: Essential Calculus: Early Transcendentals, 2nd edition, by Stewart.
Here, you can find more information about the course of Calculus 1 at USF.
Announcements
- Here or at the bottom of this page you can find the course documents (notes, quiz, homework, etc.) of Calculus 1 - Fall 2017.
- The third homework is due on Wednesday April 4 at the beginning of the class.
- On Friday April 6 I am holding a review session for the third test from 2pm to 6pm in NES 104.
Logbook:
- 01/08/18: Presentation of the course. First day attendance. Introduction to calculus. Notation. Function: definition, domain, range, different representations (graph of a function), example. Vertical line test. Exercise (find the domain of a function).
Class 1 - 1.1, 1.2Slides - Introduction to calculus - 01/10/18: Composition of two functions. Limit of a function (idea, notation, intuitive way with table of values, examples, definition). One-sided limits (left-hand, right-hand limit). Limit Laws.
Class 2 - 1.3Video - 1.3 - 01/17/18: Quiz 1. Comments on quiz 1. Direct substitution property. Classical examples: rational function $``\frac{0}{0}"$, conjugate, special limit $\lim_{x\to 0} \frac{\sin(x)}{x}=1$. Examples.
Class 3 - 1.4Video - 1.4 - 01/22/18: Limit of a piecewise function. Squeeze theorem. Continuity: definition, kinds of discontinuity (removable, jump, infinite) and examples. Theorems about continuity (sum, difference, product, quotient, composition of continuous functions).
Class 4 - 1.5Video - 1.5 - 01/24/18: Quiz 2. Exercise about continuity (piecewise function with a parameter). Intermediate Value Theorem. Exercise about IVT (show that an equation has at least one solution in an interval). Limits of the form $``\frac{1}{0}"$ (examples).
- 01/29/18: Limits at $\infty$ and $-\infty$. "Operations" with $\infty$ and $-\infty$ (indeterminate forms). Limit at infinity of a rational function $\frac{P(x)}{Q(x)}$ (three cases with examples: $\deg(P)=\deg(Q)$, $\deg(P)<\deg(Q)$ and $\deg(P)>\deg(Q)$). Definition of vertical and horizontal asymptote. Introduction to derivatives (position functon and istantaneous velocity).
Class 5/6 - 1.6Video - 1.6 - 01/31/18: Quiz 3. Comments on Quiz 3. Istantaneous velocity, slope of the tangent line to a graph at a point, instantaeous rate of change. Definition of derivative and derivative function (examples). Definition of differentiability. Examples: $f(x)=\sqrt{x}, |x|$. Theorem: differentiability implies continuity. How a function can fail to be differentiable.
Class 6/7 - 2.1Class 7 - 2.2Video - 2.1Video - 2.2 - 02/01/18: Review session Test 1.
- 02/05/18: Logic implication . Proof of the theorem: differentiability implies continuity. Differentiation rules: derivative of a constant function, power rule (ex: $x,x^2,x^3,x^{-1}=\frac{1}{x},x^{\frac{1}{2}}=\sqrt{x}$), derivatives of sine and cosine, sum rule (with proof), product rule (with proof). Examples.
Slides - Logic implicationClass 8 - 2.3, 2.4Video - 2.3Video - 2.4 - 02/07/18: Quotient rule and examples (derivative of $\tan(x)$). Chain rule (Lagrange and Leibniz notation) and examples. Introduction to implicit differentiation.
Class 9 - 2.5Video - 2.5 - 02/12/18: When do we need chain rule?: $\cos(7x)$, $7\cos(x)$,$\sin(x)\tan(x)$,$\sin(\tan(x))$,$(\sqrt[3]{x})^2$,$\frac{1}{\cos(x)}.$ Other examples of derivatives ($\sqrt{\tan(x^2)},\frac{x\cos(2x)}{\sin(x)}$). Functions explicitely defined. Functions implicitely defined: example $x^2+y^2=1$ (algebraic and geometric interpretation). Problem: find the tangent line to the curve $x^2+y^2=1$ at $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. Resolution without implicit differentiation.
- 02/14/18: Quiz 4. Method of implicit differentiation (description in 4 steps). Exercise 1: find an equation of the tangent line to the curve described by the equation $y^2+xy=\cos(y)-\sin(x)$ at the point $(\frac{\pi}{2},0)$. Exercise 2: find an equation of the tangent line to the heart curve described by the equation $(x^2+y^2-1)^3=x^2y^3$ at the point $(1,1)$ (it is Valentine's day afterall!) .
Class 10 - 2.6Video - 2.6 - 02/19/18: Review of the definition of the instantaneous rate of change: with respect to time (rate of change of the position function with respect to time -> velocity; rate of change of the velocity function with respect to time -> acceleration) or other quantities (rate of change of the pressure of an object submerged in a fluid with respect to the depth of the object). Presentation of a related rate problem through steps. Examples: square with the length of the sides increasing, air pumped into a spherical ballon, ladder sliding down.
Class 11 - 2.7Video - 2.7 - 02/21/18: Quiz 5. How does an exponential function look like? How to define $2^{\sqrt{2}}$? Review: definition of exponentiation with natural numbers as exponent and properties; extension of the range of the exponent to integer, rational and real numbers. The exponential function $f(x)=a^x$ and its graph depending on the base $a$. The "most convenient" base: the Euler's constant $e$ (definition as a limit).
Class 12 - 3.1 - 02/26/18: Natural exponential function and properties. Definition of one-to-one functions, horizontal line test, examples. Definition of the inverse function of a one-to-one function, cancellation laws. The logarithmic function with base $a$, properties, cancellations laws and laws of logarithm. Notation $\log$, $\ln$. The natural logarithmic function and properties. Derivatives of logarithmic and exponential function and examples. Logarithmic differentiation ($f(x)=x^x$). Proof of the derivative of $e^x$ with logarithmic differentiation.
Class 13 - 3.2Class 14 - 3.3Video - 3.3 - 02/28/18: Method of linear approximation for approximating the value of a function with examples. Review session Test 2.
Class 15 - 2.8 - 03/01/18: Review session Test 2.
- 03/05/18: Inverse trigonometric functions: $\sin^{-1}$ (or $\arcsin$), $\cos^{-1}$ (or $\arccos$), $\tan^{-1}$ (or $\arctan$). Domain and range, graph, cancellation equations. Examples of simplification. Derivatives (proof of the derivative of $\sin^{-1}(x)$ and $\tan^1(x)$).
Class 16 - 3.5Video - 3.5 - 03/07/18: Indeterminate forms. Statement of l'Hospital's rule and idea behind the proof. Examples. The case $0\cdot \infty$ and $0^0, 1^{\infty}, \infty^0$ with examples.
Class 17 - 3.7Video - 3.7 - 03/19/18: Definition of absolute/local maximum/minimum value. Geometrical interpretation. Examples. Extreme Value Theorem and remarks. Fermat's theorem, proof and remarks. Definition of critical numbers and example. Generalized version of Fermat's theorem.
Class 18 - 4.1 - 03/21/18: Quiz 6. Quick review of the previous class. Closed interval method and example. Rolle's theorem and Mean Value Theorem. Interpretation of MVT with a position function.
Class 19 - 4.2Video - 4.2 - 03/26/18: Review statement of the Mean Value Theorem and example. Proposition: $f'(x)=0$ on $(a,b)$ then $f$ constant on $(a,b)$ (with proof). Definition of strictly increasing/decreasing. Increasing/decreasing test (with proof). First derivative test. Example. Definition of concave up/down and inflection point.
Class 20 - 4.3Video - 4.3 Part 1Video - 4.3 Part 2 - 03/28/18: Quiz 7. Comments on the bonus question of the quiz. Review of the definition of concave up/down and inflection point. Concavity test (idea of the proof). Example. Beginning of the sketching of the graph of $f(x)=xe^{-x}$.
Class 21 - 4.4 - 04/02/18: Sketching of the graph of $f(x)=xe^{-x}$. Exercise: getting information about a function $f$ from the graph of its derivative. Optimization problem.
Video - 4.5 Part 1Video - 4.5 Part 1 - 04/04/18: Quiz 8. Comments on Quiz 8. In-class review session Test 3.
- 04/06/18: Review session Test 3.
- 04/09/18: Definition of antiderivative. The most general antiderivative. Table of antiderivatives. Exercises: Find the antiderivative of a function that satisfies a given condition - find the position function given the acceleration function, the initial velocity and the initial displacement.
Class 23 - 4.7 - 04/11/18: The area problem: easy cases ($f(x)=c$, $f(x)=\frac{1}{2}x$); $f(x)=x^2$ on $[0,1]$ and approximation through rectangles; general case of a positive function $\rightarrow$ Riemann sums (left, right, midpoint). Definition of the definite integral as the limit of Riemann sums.
Class 24 - 5.1, 5.2 - 04/16/18: Review of the definition of the definite integral. Sum notation: $\Sigma$. Computation of $\int_{0}^{1}x^2\, dx$ with the definition (limit of a Riemann sum). The definite integral of a negative function. Exercises: approximation of an integral with Riemann sums; computation of the integral of a function whose graph is made by lines and semicircles.
- 04/18/18: Quiz 9. Comments on Quiz 9. Properties of the definite integral and exercise. The integral function $g(x)=\int_{a}^{x}f(t) dt$. The integral function when $f(t)=c$ and $f(t)=mt$. Statement of the Fundamental Theorem of Calculus (Part 1 and Part 2).
- 04/23/18: Fundamental Theorem of Calulus: proof of Part 2 using Part 1. Exercises: evaluation of definites integrals, derivative of an integral function (with chain rule). Defintion and properties of the indefinite integral. Example.
Class 25 - 5.3, 5.4 - 04/24/18: Review Integrals.
- 04/25/18: The last quiz. Comments on the quiz. In-class review session Final Exam.
- 04/26/18: Review session Final Exam.
Course documents:
- Notes: ReviewLimit of a functionCalculating limitsContinuityLimits involving infinityIntroduction to derivativesDerivative as a functionDifferentiation rulesChain ruleImplicit differentiationRelated ratesExponential functionsInverse function and logarithmsDerivatives of $\log_a(x)$ and $a^x$Linear approximationInverse trig. functionsL'Hospital's ruleMaximum and minimum valuesMean Value TheoremDerivatives and shapes of the graphsCurve SketchingAntiderivativesRiemann sums and the definite integralFundamental Theorem of Calculus
- Slides: Introduction to calculusLogic implication
- Videos (by Robert Connelly): 1.31.41.51.62.12.22.32.42.52.62.73.33.53.74.24.3 - Part 14.3 - Part 24.5 - Part 14.5 - Part 2
- Sheets: DerivativesSteps - Impl. diff., related rates, log. diff.
- Quizzes: Quiz 1Q1 solutionQuiz 2Q2 solutionQuiz 3Q3 solutionQuiz 4Q4 solutionQuiz 5Q5 solutionQuiz 6Q6 solutionQuiz 7Q7 solutionQuiz 8Q8 solutionQuiz 9Quiz 10
- Homework: Homework 1HW 1 solutionHomework 2HW 2 solutionHomework 3HW 3 solution
- Reviews: Review Test 1In class review Test 2Review Test 2In class review Test 3Review Test 3Review integralsIn class review Final ExamReview Final Exam